Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
Cost: 6720
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\left(x \cdot y + x\right) + y \]
↓
\[x + \mathsf{fma}\left(x, y, y\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))
↓
(FPCore (x y) :precision binary64 (+ x (fma x y y)))
double code(double x, double y) {
	return ((x * y) + x) + y;
}
↓
double code(double x, double y) {
	return x + fma(x, y, y);
}

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end
↓
function code(x, y)
	return Float64(x + fma(x, y, y))
end

code[x_, y_] := N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision]
↓
code[x_, y_] := N[(x + N[(x * y + y), $MachinePrecision]), $MachinePrecision]

\left(x \cdot y + x\right) + y
↓
x + \mathsf{fma}\left(x, y, y\right)

Error

Derivation

  1. Initial program 0.0

    \[\left(x \cdot y + x\right) + y \]

  2. Simplified0.0

    \[\leadsto \color{blue}{x + \mathsf{fma}\left(x, y, y\right)} \]
    Proof
    (+.f64 x (fma.f64 x y y)): 0 points increase in error, 0 points decrease in error
    (+.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) y))): 2 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x (*.f64 x y)) y)): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x y) x)) y): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

    \[\leadsto x + \mathsf{fma}\left(x, y, y\right) \]

Alternatives

Alternative 1
Error8.9
Cost848
\[\begin{array}{l} t_0 := y \cdot \left(x + 1\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-201}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error8.5
Cost716
\[\begin{array}{l} t_0 := x \cdot \left(y + 1\right)\\ t_1 := y \cdot \left(x + 1\right)\\ \mathbf{if}\;y \leq 2.5 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error16.0
Cost592
\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-201}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-170}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 4
Error18.2
Cost460
\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-201}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-170}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
Alternative 5
Error0.0
Cost448
\[y + \left(x + x \cdot y\right) \]
Alternative 6
Error35.9
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))